Single-locus polymorphism in a heterogeneous two-deme model

Environmental heterogeneity has long been considered a likely explanation for the high levels of genetic variation found in most natural populations: selection in a spatially heterogeneous environment can maintain more variation. While this theoretical result has been extensively studied in models with limited parameters (e.g., two alleles, fixed gene flow, and particular selection schemes), the effect of spatial heterogeneity is poorly understood for models with a wider range of parameters (e.g., multiple alleles, different levels of gene flow, and more general selection schemes). We have compared the volume of fitness space that maintains variation in a single-deme model to the volume in a two-deme model for multiple alleles, random selection schemes, and various levels of migration. Furthermore, equilibrium allele-frequency vectors were examined to see if particular patterns of variation are more prevalent than first expected. The two-deme model maintains variation for substantially larger volumes of fitness space with lower heterozygote fitness than the single-deme model. This result implies that selection schemes in the two-deme model can have a wider range of fitness patterns while still maintaining variation. The equilibrium allele-frequency patterns emerging from the two-deme model are more variable and strongly influenced by gene flow.     

Effects of environmental heterogeneity on victim-exploiter coevolution

We study victim-exploiter coevolution in a spatially heterogeneous island model. In each species, fitness consequences of between-species interactions are controlled by a single haploid diallelic locus. Our emphasis is on the conditions for the maintenance of genetic variation, the dynamic patterns observed, the extent of local adaptation and genetic differentiation between different demes, and on how different parameters (such as the strength and heterogeneity in selection, migration rates, and the number of sites) affect the dynamic and static behavior of the system. We show that under spatially homogeneous selection the maintenance of genetic variation is possible through asynchronous nonlinear dynamics where the allele frequencies in a majority of demes quickly synchronize but the rest do not. Spatially heterogeneous selection can maintain genetic variation even if migration rates are maximal. This happens in an oscillatory way. Genetic variation is most likely to be maintained at high levels if the heterogeneity in selection is large. If there are some restrictions on migration, genetic variation can be maintained at a stable equilibrium. This behavior is most likely at intermediate migration rates. In this case, the system can exhibit high spatial subdivision as measured by F(ST) values but relatively low local adaptation.     

Evolution and polymorphism in the multilocus Levene model with no or weak epistasis

Evolution and the maintenance of polymorphism under the multilocus Levene model with soft selection are studied. The number of loci and alleles, the number of demes, the linkage map, and the degree of dominance are arbitrary, but epistasis is absent or weak. We prove that, without epistasis and under mild, generic conditions, every trajectory converges to a stationary point in linkage equilibrium. Consequently, the equilibrium and stability structure can be determined by investigating the much simpler gene-frequency dynamics on the linkage-equilibrium manifold. For a haploid species an analogous result is shown. For weak epistasis, global convergence to quasi-linkage equilibrium is established. As an application, the maintenance of multilocus polymorphism is explored if the degree of dominance is intermediate at every locus and epistasis is absent or weak. If there are at least two demes, then arbitrarily many multiallelic loci can be maintained polymorphic at a globally asymptotically stable equilibrium. Because this holds for an open set of parameters, such equilibria are structurally stable. If the degree of dominance is not only intermediate but also deme independent, and loci are diallelic, an open set of parameters yielding an internal equilibrium exists only if the number of loci is strictly less than the number of demes. Otherwise, a fully polymorphic equilibrium exists only nongenerically, and if it exists, it consists of a manifold of equilibria. Its dimension is determined. In the absence of genotype-by-environment interaction, however, a manifold of equilibria occurs for an open set of parameters. In this case, the equilibrium structure is not robust to small deviations from no genotype-by-environment interaction. In a quantitative-genetic setting, the assumptions of no epistasis and intermediate dominance are equivalent to assuming that in every deme directional selection acts on a trait that is determined additively, i.e., by nonepistatic loci with dominance. Some of our results are exemplified in this quantitative-genetic context.     

Multilocus selection in subdivided populations II. Maintenance of polymorphism under weak or strong migration

The potential of maintaining multilocus polymorphism by migration-selection balance is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, selection may vary across demes, and loci are multiallelic. It is shown that if migration and recombination are strong relative to selection, then with weak or no epistasis and intermediate dominance at every locus and in every deme, arbitrarily many alleles can be maintained at arbitrarily many loci at a stable equilibrium. If migration is weak relative to selection and recombination, then with weak or no epistasis and intermediate dominance at every locus and in every deme, as many alleles as there are demes can be maintained at arbitrarily many loci at equilibrium. In both cases open sets of such parameter combinations are constructed, thus the results are robust with respect to small, but arbitrary, perturbations in the parameters. For weak migration, the number of demes is, in fact, a generic upper bound to the number of alleles that can be maintained at any locus. Thus, several scenarios are identified under which multilocus polymorphism can be maintained by migration-selection balance when this is impossible in a panmictic population.      

Clines with partial panmixia

In spatially distributed populations, global panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into single-locus clines maintained by migration and selection is investigated. In a diallelic, two-deme model without dominance, partial panmixia can increase or decrease both the polymorphic area in the plane of the migration rates and the equilibrium gene-frequency difference between the two demes. For multiple alleles, under the assumptions that the number of demes is large and both migration and selection are arbitrary but weak, a system of integro-partial differential equations is derived. For two alleles with conservative migration, (i) a Lyapunov functional is found, suggesting generic global convergence of the gene frequency; (ii) conditions for the stability or instability of the fixation states, and hence for a protected polymorphism, are obtained; and (iii) a variational representation of the minimal selection-migration ratio λ₀ (the principal eigenvalue of the linearized system) for protection from loss is used to prove that λ₀ is an increasing function of the panmictic rate and to deduce the effect on λ₀ of changes in selection and migration. The unidimensional step-environment with uniform population density, homogeneous, isotropic migration, and no dominance is examined in detail: An explicit characteristic equation is derived for λ₀; bounds on λ₀ are established; and λ₀ is approximated in four limiting cases. An explicit formula is also deduced for the globally asymptotically stable cline in an unbounded habitat with a symmetric environment; partial panmixia maintains some polymorphism even as the distance from the center of the cline tends to infinity.     

Between migration load and evolutionary rescue: dispersal,adaptation and the response of spatially structured populations to environmental change

The evolutionary potential of populations is mainly determined by population size and available genetic variance. However, the adaptability of spatially structured populations may also be affected by dispersal: positively by spreading beneficial mutations across sub-populations, but negatively by moving locally adapted alleles between demes. We develop an individual-based, two-patch, allelic model to investigate the balance between these opposing effects on a population''s evolutionary response to rapid climate change. Individual fitness is controlled by two polygenic traits coding for local adaptation either to the environment or to climate. Under conditions of selection that favour the evolution of a generalist phenotype (i.e. weak divergent selection between patches) dispersal has an overall positive effect on the persistence of the population. However, when selection favours locally adapted specialists, the beneficial effects of dispersal outweigh the associated increase in maladaptation for a narrow range of parameter space only (intermediate selection strength and low linkage among loci), where the spread of beneficial climate alleles is not strongly hampered by selection against non-specialists. Given that local selection across heterogeneous and fragmented landscapes is common, the complex effect of dispersal that we describe will play an important role in determining the evolutionary dynamics of many species under rapidly changing climate.     

Polymorphism in the two-locus Levene model with nonepistatic directional selection

For the Levene model with soft selection in two demes, the maintenance of polymorphism at two diallelic loci is studied. Selection is nonepistatic and dominance is intermediate. Thus, there is directional selection in every deme and at every locus. We assume that selection is in opposite directions in the two demes because otherwise no polymorphism is possible. If at one locus there is no dominance, then a complete analysis of the dynamical and equilibrium properties is performed. In particular, a simple necessary and sufficient condition for the existence of an internal equilibrium and sufficient conditions for global asymptotic stability are obtained. These results are extended to deme-independent degree of dominance at one locus. A perturbation analysis establishes structural stability within the full parameter space. In the absence of genotype-environment interaction, which requires deme-independent dominance at both loci, nongeneric equilibrium behavior occurs, and the introduction of arbitrarily small genotype-environment interaction changes the equilibrium structure and may destroy stable polymorphism. The volume of the parameter space for which a (stable) two-locus polymorphism is maintained is computed numerically. It is investigated how this volume depends on the strength of selection and on the dominance relations. If the favorable allele is (partially) dominant in its deme, more than 20% of all parameter combinations lead to a globally asymptotically stable, fully polymorphic equilibrium.     

Polymorphism in multiallelic migration–selection models with dominance

The evolution of the gene frequencies at a single multiallelic locus under the joint action of migration and viability selection with dominance is investigated. The monoecious, diploid population is subdivided into finitely many panmictic colonies that exchange adult migrants independently of genotype. Underdominance and overdominance are excluded. If the degree of dominance is deme independent for every pair of alleles, then under the Levene model, the qualitative evolution of the gene frequencies (i.e., the existence and stability of the equilibria) is the same as without dominance. In particular: (i) the number of demes is a generic upper bound on the number of alleles present at equilibrium; (ii) there exists exactly one stable equilibrium, and it is globally attracting; and (iii) if there exists an internal equilibrium, it is globally asymptotically stable. Analytic examples demonstrate that if either the Levene model does not apply or the degree of dominance is deme dependent, then the above results can fail. A complete global analysis of weak migration and weak selection on a recessive allele in two demes is presented.     

Environmental variation, fluctuating selection and genetic drift in subdivided populations

Although there have many studies of the population genetical consequences of environmental variation, little is known about the combined effects of genetic drift and fluctuating selection in structured populations. Here we use diffusion theory to investigate the effects of temporally and spatially varying selection on a population of haploid individuals subdivided into a large number of demes. Using a perturbation method for processes with multiple time scales, we show that as the number of demes tends to infinity, the overall frequency converges to a diffusion process that is also the diffusion approximation for a finite, panmictic population subject to temporally fluctuating selection. We find that the coefficients of this process have a complicated dependence on deme size and migration rate, and that changes in these demographic parameters can determine both the balance between the dispersive and stabilizing effects of environmental variation and whether selection favors alleles with lower or higher fitness variance.     

Genetic Variation in a Heterogeneous Environment. II. Temporal Heterogeneity and Directional Selection

The maintenance of genetic variation is investigated in a finite population where selection at an autosomal locus with two alleles varies temporally between two environments and the heterozygote has an intermediate fitness value. When there is additive gene action and equal selection in both environments, the autocorrelation between subsequent environments must be negative for more maintenance of genetic variation than for neutrality. The maximum maintenance occurs when there is equal selection in the two environments and the autocorrelation approaches ^-1.0 (for a stochastic model), or when there is short repeating cycle such as one related to seasons. Also comparison of the effects of stochastic variation in selection in finite and infinite populations is made by using Monte Carlo simulation. One situation was found where temporal environmental variation maintains genetic variation very effectively even in a small population and that is when there is evolution of dominance, i.e., the heterozygote is closer in fitness to the favored homozygote than the other homozygote. An important conclusion is that in a finite population genetic tracing of environmental change, particularly when there is a positive autocorrelation between environments or a long environmental cycle, leads to an increased loss of genetic variation making such a response undesirable in the long term, a result different from that in infinite populations.     

Protected polymorphism and evolutionary stability in pleiotropic models with trait-specific dominance

When alleles have pleiotropic effects on a number of quantitative traits, the degree of dominance between a pair of alleles can be different for each trait. Such trait-specific dominance has been studied previously in models for the maintenance of genetic variation by antagonistic effects of an allele on two fitness components. By generalizing these models to an arbitrary number of fitness components or other phenotypic traits with different degrees of dominance, I show that genetic polymorphism is generally impossible without antagonistic fitness effects of different traits and without trait-specific dominance. I also investigate dominance and pleiotropy from a more long-term evolutionary perspective, allowing for the study of general ecological scenarios, and I discuss the effects of trait-specific dominance on evolutionary stability criteria. When selection is mainly directional and only trait-specific dominance and antagonism cause the emergence of polymorphism, then these polymorphisms can be overtaken by single mutants again, such that they are probably short-lived on an evolutionary time scale. Near evolutionarily singular points where directional selection is absent, trait-specific dominance and overdominance facilitate the emergence of polymorphism and cause evolutionary divergence in some cases. An important outcome of these models is that trait-specific dominance allows for the emergence of genetic polymorphisms without a selective disadvantage for heterozygotes. This removes the scope for the evolution of assortative mate choice and affects dominance modification. Sympatric speciation by disruptive ecological selection requires this heterozygote disadvantage in order to evolve, and therefore it becomes less plausible if the emergence of genetic polymorphism usually occurs via trait-specific dominance and antagonistic effects.     

The number of equilibria in the diallelic Levene model with multiple demes

The Levene model is the simplest mathematical model to describe the evolution of gene frequencies in spatially subdivided populations. It provides insight into how locally varying selection promotes a population’s genetic diversity. Despite its simplicity, interesting problems have remained unsolved even in the diallelic case.In this paper we answer an open problem by establishing that for two alleles at one locus and J demes, up to 2J−1 polymorphic equilibria may coexist. We first present a proof for the case of stable monomorphisms and then show that the result also holds for protected alleles. These findings allow us to prove that any odd number (up to 2J−1) of equilibria is possible, before we extend the proof to even numbers. We conclude with some numerical results and show that for J>2, the proportion of parameter space affording this maximum is extremely small.     

Consequences of genetic linkage for the maintenance of sexually antagonistic polymorphism in hermaphrodites

When selection differs between males and females, pleiotropic effects among genes expressed by both sexes can result in sexually antagonistic selection (SA), where beneficial alleles for one sex are deleterious for the other. For hermaphrodites, alleles with opposing fitness effects through each sex function represent analogous genetic constraints on fitness. Recent theory based on single‐locus models predicts that the maintenance of SA genetic variation should be greatly reduced in partially selfing populations. However, selfing also reduces the effective rate of recombination, which should facilitate selection on linked allelic combinations and expand opportunities for balancing selection in a multilocus context. Here, I develop a two‐locus model of SA selection for simultaneous hermaphrodites, and explore the joint influence of linkage, self‐fertilization, and dominance on the maintainance of SA polymorphism. I find that the effective reduction in recombination caused by selfing significantly expands the parameter space where SA polymorphism can be maintained relative to single‐locus models. In particular, linkage facilitates the invasion of male‐beneficial alleles, partially compensating for the “female‐bias” in the net direction of selection created by selfing. I discuss the implications of accounting for linkage among SA loci for the maintenance of SA genetic variation and mixed mating systems in hermaphrodites.     

Gynodioecy in structured populations: understanding fine-scale sex ratio variation in Beta vulgaris ssp. maritima

Natural selection, random processes and gene flow are known to generate sex ratio variations among sexually polymorphic plant populations. In gynodioecious species, in which hermaphrodites and females coexist, the relative effect of these processes on the maintenance of sex polymorphism is still up for debate. The aim of this study was to document sex ratio and cytonuclear genetic variation at a very local scale in wind-pollinated gynodioecious Beta vulgaris ssp. maritima and attempt to elucidate which processes explained the observed variation. The study sites were characterized by geographically distinct patches of individuals and appeared to be dynamic entities, with recurrent establishment of distinct haplotypes through independent founder events. Along with substantial variation in sex ratio and unexpectedly low gene flow within study sites, our results showed a high genetic differentiation among a mosaic of genetically distinct demes, with isolation by distance or abrupt genetic discontinuities taking place within a few tens of metres. Overall, random founder events with restricted gene flow could be primary determinants of sex structure, by promoting the clumping of sex-determining genes. Such high levels of sex structure provide a landscape for differential selection acting on sex-determining genes, which could modify the conditions of maintenance of gynodioecy in structured populations.     

Evolution of altruism under group selection in large and small populations in fluctuating environments

A continuous, graded form of group selection which does not involve extinction of demes can effectively oppose selection on the individual level against an altruistic allele under fluctuating environments in infinitely large demes among which uniform mixing occurs every generation. Although group selection cannot alter the conditions necessary for the initial increase of altruistic alleles, group selection can significantly influence the stationary distribution of gene frequency which is attained once stochastic forces have allowed theirintroduction. Drift is a more effective source of variation than fluctuations in selection when the variance in selection is moderate to small. High numbers of demes promote polymorphism under both graded group selection and extinction group selection.     

Influence of dominance, leptokurtosis and pleiotropy of deleterious mutations on quantitative genetic variation at mutation-selection balance

In models of maintenance of genetic variance (V (G)) it has often been assumed that mutant alleles act additively. However, experimental data show that the dominance coefficient varies among mutant alleles and those of large effect tend to be recessive. On the basis of empirical knowledge of mutations, a joint-effect model of pleiotropic and real stabilizing selection that includes dominance is constructed and analyzed. It is shown that dominance can dramatically alter the prediction of equilibrium V (G). Analysis indicates that for the situations where mutations are more recessive for fitness than for a quantitative trait, as supported by the available data, the joint-effect model predicts a significantly higher V (G) than does an additive model. Importantly, for what seem to be realistic distributions of mutational effects (i.e., many mutants may not affect the quantitative trait substantially but are likely to affect fitness), the observed high levels of genetic variation in the quantitative trait under strong apparent stabilizing selection can be generated. This investigation supports the hypothesis that most V (G) comes from the alleles nearly neutral for fitness in heterozygotes while apparent stabilizing selection is contributed mainly by the alleles of large effect on the quantitative trait. Thus considerations of dominance coefficients of mutations lend further support to our previous conclusion that mutation-selection balance is a plausible mechanism of the maintenance of the genetic variance in natural populations.     

Major histocompatibility complex variation in insular populations of the Egyptian vulture: inferences about the roles of genetic drift and selection

Insular populations have attracted the attention of evolutionary biologists because of their morphological and ecological peculiarities with respect to their mainland counterparts. Founder effects and genetic drift are known to distribute neutral genetic variability in these demes. However, elucidating whether these evolutionary forces have also shaped adaptive variation is crucial to evaluate the real impact of reduced genetic variation in small populations. Genes of the major histocompatibility complex (MHC) are classical examples of evolutionarily relevant loci because of their well-known role in pathogen confrontation and clearance. In this study, we aim to disentangle the partial roles of genetic drift and natural selection in the spatial distribution of MHC variation in insular populations. To this end, we integrate the study of neutral (22 microsatellites and one mtDNA locus) and MHC class II variation in one mainland (Iberia) and two insular populations (Fuerteventura and Menorca) of the endangered Egyptian vulture (Neophron percnopterus). Overall, the distribution of the frequencies of individual MHC alleles (n=17 alleles from two class II B loci) does not significantly depart from neutral expectations, which indicates a prominent role for genetic drift over selection. However, our results point towards an interesting co-evolution of gene duplicates that maintains different pairs of divergent alleles in strong linkage disequilibrium on islands. We hypothesize that the co-evolution of genes may counteract the loss of genetic diversity in insular demes, maximize antigen recognition capabilities when gene diversity is reduced, and promote the co-segregation of the most efficient allele combinations to cope with local pathogen communities.     

Antagonistic pleiotropy in species with separate sexes,and the maintenance of genetic variation in life‐history traits and fitness

Antagonistic pleiotropy (AP)—where alleles of a gene increase some components of fitness at a cost to others—can generate balancing selection, and contribute to the maintenance of genetic variation in fitness traits, such as survival, fecundity, fertility, and mate competition. Previous theory suggests that AP is unlikely to maintain variation unless antagonistic selection is strong, or AP alleles exhibit pronounced differences in genetic dominance between the affected traits. We show that conditions for balancing selection under AP expand under the likely scenario that the strength of selection on each fitness component differs between the sexes. Our model also predicts that the vast majority of balanced polymorphisms have sexually antagonistic effects on total fitness, despite the absence of sexual antagonism for individual fitness components. We conclude that AP polymorphisms are less difficult to maintain than predicted by prior theory, even under our conservative assumption that selection on components of fitness is universally sexually concordant. We discuss implications for the maintenance of genetic variation, and for inferences of sexual antagonism that are based on sex‐specific phenotypic selection estimates—many of which are based on single fitness components.     

Hitchhiking effect of a beneficial mutation spreading in a subdivided population

A central problem in population genetics is to detect and analyze positive natural selection by which beneficial mutations are driven to fixation. The hitchhiking effect of a rapidly spreading beneficial mutation, which results in local removal of standing genetic variation, allows such an analysis using DNA sequence polymorphism. However, the current mathematical theory that predicts the pattern of genetic hitchhiking relies on the assumption that a beneficial mutation increases to a high frequency in a single random-mating population, which is certainly violated in reality. Individuals in natural populations are distributed over a geographic space. The spread of a beneficial allele can be delayed by limited migration of individuals over the space and its hitchhiking effect can also be affected. To study this effect of geographic structure on genetic hitchhiking, we analyze a simple model of directional selection in a subdivided population. In contrast to previous studies on hitchhiking in subdivided populations, we mainly investigate the range of sufficiently high migration rates that would homogenize genetic variation at neutral loci. We provide a heuristic mathematical analysis that describes how the genealogical structure at a neutral locus linked to the locus under selection is expected to change in a population divided into two demes. Our results indicate that the overall strength of genetic hitchhiking--the degree to which expected heterozygosity decreases--is diminished by population subdivision, mainly because opportunity for the breakdown of hitchhiking by recombination increases as the spread of the beneficial mutation across demes is delayed when migration rate is much smaller than the strength of selection. Furthermore, the amount of genetic variation after a selective sweep is expected to be unequal over demes: a greater reduction in expected heterozygosity occurs in the subpopulation from which the beneficial mutation originates than in its neighboring subpopulations. This raises a possibility of detecting a "hidden" geographic structure of population by carefully analyzing the pattern of a selective sweep.     

GEOGRAPHIC VARIATION, FREQUENCY-DEPENDENT SELECTION, AND THE MAINTENANCE OF A FEMALE-LIMITED POLYMORPHISM

A central problem in evolutionary biology is to understand how spatial and temporal variation in selection maintain genetic variation within and among populations. Brown anole lizards ( Anolis sagrei ) exhibit a dorsal pattern polymorphism that is expressed only in females, which occur in "diamond,""bar," and intermediate "diamond-bar" morphs. To understand the inheritance of this polymorphism, we conducted a captive breeding study that refuted several single-locus models and supported a two-locus mode of inheritance. To describe geographic variation in morph frequencies, we surveyed 13 populations from two major islands in The Bahamas. Morph frequencies differed substantially between major islands but were highly congruent within each island. Finally, we measured viability selection on each island to test two hypotheses regarding the maintenance of the polymorphism: (1) that spatial variation in selection maintains variation in morph frequencies between islands, and (2) that temporal variation in selection across years maintains variation within islands. Although bar females had relatively lower survival where they were rare, our data do not otherwise suggest that selection varies spatially between islands. However, diamond-bar females were subject to positive frequency-dependent selection across years, and the relative fitness of bar and diamond females alternated across years. We propose that this polymorphism is maintained by temporal variation in selection coupled with the sheltering of alleles via a two-locus inheritance pattern and sex-limited expression.